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Total of all digits written

BrainteasersEasy

The integers from $1$ to $250$ are written out in ordinary decimal notation with no leading zeros, and every digit written is added to a running total. What is the final total?

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Hint

Split the range into blocks where the digit sums are easy: single digits, $2$ -digit numbers, then the hundreds.

Solution

The numbers $1$ to $9$ contribute $45$ , and the numbers $10$ to $99$ contribute $10 \times 45 + 9 \times 45 = 855$ , because each tens digit from $1$ to $9$ appears $10$ times and each units digit appears $9$ times.
The numbers $100$ to $199$ contribute $100 \times 1$ from the hundreds digit plus $900$ from the last $2$ digits, which is $1000$ .
The numbers $200$ to $250$ contribute $51 \times 2 = 102$ from the hundreds digit, $100$ from the tens digits of $200$ to $249$ , $5 \times 45 = 225$ from their units digits, and $5$ from the tens digit of $250$ , which is $432$ in all.
Adding the blocks gives $45 + 855 + 1000 + 432 = 2332$ .

Answer: $2332$

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